There have been several Simpsons episodes involving the mafia. Therefore there have been several movie references to the Godfather movies. Joe Mantegna who gives the voice to Fat Tony Williams plays Joey Zosa in The Godfather Part 3.
Season 3 - 8F06 - Lisa's Pony

In the Godfather Part 1, does a movie director wake up with the head of his $600000 horse in his bed.

  Lisa screams when she finds a pony in her bed.
Season 10 - AABF05 - Mayored to the Mob
A guy thanking Don Corleone.  

Homer giving Mayor Quimby the "kiss of death".

Michael Corleone killing the police commissioner and another guy before he drops the gun and leaves the restaurant.  

Fat Tony beating Quimby with a base ball bath, before he drops it on the floor and says: "What? What did I do?"

8F03 - Bart the Murderer
Michael Corleone as the leader of the Corleone Family.  

Bart Simpsons as the leader of the Legal Business Man's Social Club.

Season 13 - DABF14 - The Sweetest Apu
Some people singing a song in the wedding in the Godfather Part 1.  

Homer singing the exact same song in Apu's wedding.

Season 14 - EABF04 - Strong Arms of the Ma
Sonny takes revenge over his sister's husband after he has beaten her up.   Marge takes revenge over the guy that robbed her, by beating him up just like Sonny does in the Godfather part 1.
Season 16 - FABF20 - All's Fair in Oven War
Sonny's life ends quickly in Godfather Part 1.   Cletus gets revenge over the guy that stole his fiancé.

Season 3 - 8F03 - Bart the Murderer

Similar Quote
Don Corleone:
What have I ever done to make you treat me so disrespectfully?
  Similar Quote
The Big Cheese:
What have I done to deserve this tasteless Manhatten?
Thanks to Springfield Paradise for some of these framegrabs.